chi_square

Chi-Square Test
A fundamental problem in genetics is determining
whether the experimentally determined data fits the
results expected from theory (i.e. Mendel’s laws as
expressed in the Punnett square).

Chi-Square Test
How can you tell if an observed set of offspring
counts is legitimately the result of a given underlying
simple ratio?
For example, you do a cross and see 290 purple flowers
and 110 white flowers in the offspring.
This is pretty close to a 3/4 : 1/4 ratio, but how do you
formally define "pretty close"? What about 250:150?

Goodness of Fit
Mendel had no way of solving this problem. Shortly
after the rediscovery of his work in 1900, Karl
Pearson and R.A. Fisher developed the “chi-square”
test for this purpose.
The chi-square test is a “goodness of fit” test:
it answers the question of how well do experimental data
fit expectations.

Goodness of Fit
We start with a theory for how the offspring will be
distributed: the “null hypothesis”.
We will discuss the offspring of a self-pollination of a
heterozygote.
The null hypothesis is that the offspring will appear in
a ratio of 3/4 dominant to 1/4 recessive.

Formula
First determine the
number of each
phenotype that have
been observed and how
many would be
expected given basic
genetic theory.
Then calculate the chisquare
statistic using this
formula. You need to
memorize the formula!
( obs
exp)
2
exp
2

Formula
The “Χ” is the Greek letter
chi; the “∑” is a sigma; it
means to sum the following
terms for all phenotypes.
“obs” is the number of
individuals of the given
phenotype observed; “exp”
is the number of that
phenotype expected from
the null hypothesis.
Note that you must use the
number of individuals, the
counts, and NOT proportions,
ratios, or frequencies.
( obs
exp)
2
exp
2

Example
As an example, you count F2 offspring, and get 290 purple and 110 white
flowers. This is a total of 400 (290 + 110) offspring.
We expect a 3/4 : 1/4 ratio. We need to calculate the expected numbers
(you MUST use the numbers of offspring, NOT the proportion!!!); this is done
by multiplying the total offspring by the expected proportions. This we
expect 400 * 3/4 = 300 purple, and 400 * 1/4 = 100 white.

Example
Now it's just a matter of plugging into the formula:
2 = (290 - 300) 2 / 300 + (110 - 100) 2 / 100
= (-10) 2 / 300 + (10) 2 / 100
= 100 / 300 + 100 / 100
= 0.333 + 1.000
= 1.333.
This is our chi-square value: now we need to see what it means and how to
use it.

Chi-Square Distribution
Although the chi-square
distribution can be derived through
math theory, we can also get it
experimentally:
Let's say we do the same
experiment 1000 times, do the
same self-pollination of a Pp
heterozygote, which should give
the 3/4 : 1/4 ratio. For each
experiment we calculate the chisquare
value, them plot them all on
a graph.
The x-axis is the chi-square value
calculated from the formula. The
y-axis is the number of individual
experiments that got that chisquare
value.

Chi-Square Distribution, p. 2
You see that there is a range here:
if the results were perfect you get
a chi-square value of 0 (because
obs = exp). This rarely happens:
most experiments give a small chisquare
value (the hump in the
graph).

Chi-Square Distribution, p. 3
Note that all the values are
greater than 0: that's because we
squared the (obs - exp) term:
squaring always gives a nonnegative
number.

Chi-Square Distribution, p. 4
Sometimes you get really wild
results, with obs very different
from exp: the long tail on the
graph. Really odd things
occasionally do happen by chance
alone (for instance, you might win
the lottery).

The Critical Question
How do you tell a really odd but correct result from a
WRONG result? The graph is what happens with real
experiments: most of the time the results fit
expectations pretty well, but occasionally very skewed
distributions of data occur even though you performed
the experiment correctly, based on the correct theory,
The simple answer is: you can never tell for certain
that a given result is “wrong”, that the result you got
was completely impossible based on the theory you
used. All we can do is determine whether a given
result is likely or unlikely.

The Critical Question
Key point: There are 2 ways of getting a high chisquare
value: an unusual result from the correct theory,
or a result from the wrong theory. These are
indistinguishable; because of this fact, statistics is
never able to discriminate between true and false
with 100% certainty.
Using the example here, how can you tell if your 290:
110 offspring ratio really fits a 3/4 : 1/4 ratio (as
expected from selfing a heterozygote) or whether it
was the result of a mistake or accident-- a 1/2 : 1/2
ratio from a backcross for example? You can’t be
certain, but you can at least determine whether your
result is reasonable.

Reasonable
What is a “reasonable” result is subjective and arbitrary.
For most work (and for the purposes of this class), a result is
said to not differ significantly from expectations if it could
happen at least 1 time in 20. That is, if the difference between
the observed results and the expected results is small enough
that it would be seen at least 1 time in 20 over thousands of
experiments, we “fail to reject” the null hypothesis.

Reasonable
For technical reasons, we use “fail to reject” instead of
“accept”.
“1 time in 20” can be written as a probability value p = 0.05,
because 1/20 = 0.05.
Another way of putting this. If your experimental results are
worse than 95% of all similar results, they get rejected because
you may have used an incorrect null hypothesis.

Degrees of Freedom
A critical factor in using the chi-square test is the
“degrees of freedom”, which is essentially the
number of independent random variables involved.
Degrees of freedom is simply the number of classes
of offspring minus 1.
For our example, there are 2 classes of offspring:
purple and white. Thus, degrees of freedom (d.f.)
= 2 -1 = 1.

Critical Chi-Square
Critical values for chi-square are found on tables,
sorted by degrees of freedom and probability levels.
Be sure to use p = 0.05.
If your calculated chi-square value is greater than the
critical value from the table, you “reject the null
hypothesis”.
If your chi-square value is less than the critical value,
you “fail to reject” the null hypothesis (that is, you
accept that your genetic theory about the expected
ratio is correct).

Chi-Square Table

Using the Table
In our example of 290 purple to 110 white, we
calculated a chi-square value of 1.333, with 1
degree of freedom.
Looking at the table, 1 d.f. is the first row, and p =
0.05 is the sixth column. Here we find the critical chisquare
value, 3.841.
Since our calculated chi-square, 1.333, is less than the
critical value, 3.841, we “fail to reject” the null
hypothesis. Thus, an observed ratio of 290 purple to
110 white is a good fit to a 3/4 to 1/4 ratio.

Another Example: from Mendel
phenotype observed expected
proportion
round
yellow
round
green
wrinkled
yellow
wrinkled
green
expected
number
315 9/16 312.75
101 3/16 104.25
108 3/16 104.25
32 1/16 34.75
total 556 1 556

Finding the Expected Numbers
You are given the observed numbers, and you determine the
expected proportions from a Punnett square.
To get the expected numbers of offspring, first add up the
observed offspring to get the total number of offspring. In this
case, 315 + 101 + 108 + 32 = 556.
Then multiply total offspring by the expected proportion:
--expected round yellow = 9/16 * 556 = 312.75
--expected round green = 3/16 * 556 = 104.25
--expected wrinkled yellow = 3/16 * 556 = 104.25
--expected wrinkled green = 1/16 * 556 = 34.75
Note that these add up to 556, the observed total offspring.

Calculating the Chi-Square Value
Use the formula.
X 2 = (315 - 312.75)2 / 312.75
+ (101 - 104.25)2 / 104.25
+ (108 - 104.25)2 / 104.25
+ (32 - 34.75)2 / 34.75
= 0.016 + 0.101 + 0.135 + 0.218
= 0.470.
2 ( obs exp)
exp
2

D.F. and Critical Value
Degrees of freedom is 1 less than the number of
classes of offspring. Here, 4 - 1 = 3 d.f.
For 3 d.f. and p = 0.05, the critical chi-square value
is 7.815.
Since the observed chi-square (0.470) is less than the
critical value, we fail to reject the null hypothesis. We
accept Mendel’s conclusion that the observed results
for a 9/16 : 3/16 : 3/16 : 1/16 ratio.
It should be mentioned that all of Mendel’s numbers
are unreasonably accurate.

Chi-Square Table

Mendel’s Yellow vs. Green Results